Variant-Based Decidable Satisfiability in Initial Algebras with Predicates

This work proposes canonical constrained narrowing, a new symbolic reachability analysis technique applicable to topmost rewrite theories where the equational theory has the finite variant property. Our experiments suggest that canonical constrained narrowing is more efficient than both standard narrowing and the previously studied contextual narrowing. These results are relevant not only for Maude-NPA, but also for symbolically analyzing many other concurrent systems specified by means of rewrite theories.

Lower bounds are established on the computational complexity of the decision problem and on the inherent lengths of proofs for two classical decidable theories of logic: the first-order theory of the real numbers under addition, and Presburger arithmetic — the first-order theory of addition on the natural numbers. There is a fixed constant c > 0 such that for every (nondeterministic) decision procedure for determining the truth of sentences of real addition and for all sufficiently large n, there is a sentence of length n for which the decision procedure runs for more than 2cn steps. In the case of Presburger arithmetic, the corresponding bound is \({2^{{2^{cn}}}}\). These bounds apply also to the minimal lengths of proofs for any complete axiomatization in which the axioms are easily recognized.KeywordsDecision ProcedureFinite Abelian GroupScanning HeadBinary WordReal AdditionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Inspired by Boyer and Moore''s approach for generating induction schemes based on terminating function definitions, Zhang, Kapur and Krishnamoorthy introduced a cover set method for designing induction schemes for automating proofs by induction from specifications expressed as equations and conditional equations. This method has been implemented in the theorem prover Rewrite Rule Laboratory (RRL) and a proof management system Tecton built on top of em RRL, and it has been used to prove many nontrivial theorems and reason about sequential as well as parallel programs. The cover set method is based on the assumption that a function symbol is defined using a finite set of terminating (conditional or unconditional) rewrite rules. The left side of the rules are then used to design different cases of an induction scheme, and recursive calls to the function made in the right side can be used to design appropriate instantiations for generating induction hypotheses. A weakness of this method is that it relies on syntactic unification for generating an induction scheme for a conjecture. This paper goes a step further by proposing semantic analysis for generating an induction scheme for a conjecture from a cover set. We discuss the use of a decision procedure for Presburger arithmetic (quantifier-free theory of numbers with the addition operation and relational predicates , , =, ) for performing semantic analysis about numbers. The decision procedure is used to generate appropriate induction schemes for a conjecture using cover sets of function taking numbers as arguments. This extension of the cover set method automates proofs of many theorems which otherwise, require human guidance and hints. The effectiveness of the method is demonstrated using some examples which commonly arise in reasoning about specifications and programs. It is also shown how semantic analysis using a Presburger arithmetic decision procedure can be used for checking the completeness of a cover set of a function defined using operations such as check, many function definitions used in a proof of the prime factorization theorem stating that every number can be factored uniquely into prime factors, which had to be checked manually, can now be checked automatically in procedure for guiding generalization for generating conjectures and merging induction schemes is also illustrated.

We consider the first-order theory whose language has as nonlogical symbols the constant symbols 0 and 1, the binary relation symbols = and This theory of integers under addition is commonly called the 'Presburger Arithmetic' and is known to be decidable for truth [Presburger (1929), Hilbert and Bernays (1968)]. We prove here that there exists a decision procedure for this theory, involving quantifier elimination, for which there is a superexponential upper bound on the size of formula produced when all variables have been eliminated.

The initial algebra for a set functor can be constructed iteratively via a well-known transfinite chain, which converges after a regular infinite cardinal number of steps or at most three steps. We extend this result to the analogous construction of relatively initial algebras. For the dual construction of the terminal coalgebra Worrell proved that if a set functor is α-accessible, then convergence takes at most α + α steps. But until now an example demonstrating that fewer steps may be insufficient was missing. We prove that the functor of all α-small filters is such an example. We further prove that for β ≤ α the functor of all α-small β-generated filters requires precisely α + β steps and that a certain modified power-set functor requires precisely α steps. We also present an example showing that whether a terminal coalgebra exists at all does not depend solely on the object mapping of the given set functor. (This contrasts with the fact that existence of an initial algebra is equivalent to existence of a mere fixed point.)

Motivated by applications to program verification, we study a decision procedure for satisfiability in an expressive fragment of a theory of arrays, which is parameterized by the theories of the array elements. The decision procedure reduces satisfiability of a formula of the fragment to satisfiability of an equisatisfiable quantifier-free formula in the combined theory of equality with uninterpreted functions (EUF), Presburger arithmetic, and the element theories. This fragment allows a constrained use of universal quantification, so that one quantifier alternation is allowed, with some syntactic restrictions. It allows expressing, for example, that an assertion holds for all elements in a given index range, that two arrays are equal in a given range, or that an array is sorted. We demonstrate its expressiveness through applications to verification of sorting algorithms and parameterized systems. We also prove that satisfiability is undecidable for several natural extensions to the fragment. Finally, we describe our implementation in the πVC verifying compiler.

Coq is a proof assistant featuring a tactic-based interactive theorem prover. The latest incarnation comes with over 150 tactics that assist the user in developing a formal proof. These tactics range from the simple and mundane to the ‘allpowerful’. Some examples from the latter category are the omega tactic that solves a goal in Presburger arithmetic and the ring and field tactics that solve identities modulo associativity and commutativity in ring and field structures. This paper presents a new proof tactic that decides equalities and inequalities between terms over lattices. It uses a decision procedure that is a variation on Whitman’s algorithm and is implemented using a technique known as proof by reflection. We will paint the essence of the approach in broad strokes and discuss the use of certified functional programs to aid the automation of formal reasoning. This paper makes three contributions. Firstly it serves an an introduction to using proof by reflection approach Coq. This utilizes the Ltac language and two recent extensions to the Coq system: type classes and the PROGRAM extension. Secondly it gives a certified implementation of Whitman’s algorithm in Coq, with proofs of correctness and termination. Thirdly, the final product of this work is a useable proof tactic that can be applied to proof goals involving equalities and inequalities in lattice theory. The rest of the paper is organized as follows: Section 2 will give a short introduction to Coq, highlighting the concept of a proof term, and will refresh the necessary details of lattice theory. We will give a motivation for the problem in Section 3, and a Coq-based introduction to free lattices along with the decision

This paper presents two decidability results on the validity checking problem for entailments of symbolic heaps in separation logic with Presburger arithmetic and arrays. The first result is for a system with arrays and existential quantifiers. The correctness of the decision procedure is proved under the condition that sizes of arrays in the succedent are not existentially quantified. This condition is different from that proposed by Brotherston et al. in 2017 and one of them does not imply the other. The main idea is a novel translation from an entailment of symbolic heaps into a formula in Presburger arithmetic. The second result is the decidability for a system with both arrays and lists. The key idea is to extend the unroll collapse technique proposed by Berdine et al. in 2005 to arrays and arithmetic as well as double-linked lists.Comment: arXiv admin note: text overlap with arXiv:1708.06696

This paper considers the satisfiability problem of symbolic heaps in separation logic with Presburger arithmetic and inductive definitions. First the system without any restrictions is proved to be undecidable. Secondly this paper proposes some syntactic restrictions for decidability. These restrictions are identified based on a new decidable subsystem of Presburger arithmetic with inductive definitions. In the subsystem of arithmetic, every inductively defined predicate represents an eventually periodic set and can be eliminated. The proposed system is quite general as it can handle the satisfiability of the arithmetical parts of fairly complex predicates such as sorted lists and AVL trees. Finally, we prove the decidability by presenting a decision procedure for symbolic heaps with the restricted inductive definitions and arithmetic.

Research on the automatic verification of heap-manipulating programs (HMPs)--programs that manipulate unbounded linked data structures via pointers --has blossomed recently, with many different approaches all showing leaps in performance and expressiveness. A year ago, we proposed a small logic for specifying predicates about HMPs and demonstrated that an inference-rule-based decision procedure could be performance-competitive, and in many cases superior to other methods known at the time. That work, however, was a proof-of-concept, with a logic fragment too small to verify most real programs. In this work, we generalize our previous results to be practically useful: we allow the data in heap nodes to be mutable, we allow more than a single pointer field, and we add new primitives needed to verify cyclic structures. Each of these extensions necessitates new or changed inference rules, with the concomitant changes to the proofs and decision procedure. Yet, our new decision procedure, with the more general logic, actually runs as fast as our previous results. With these generalizations, we can automatically verify many more HMP examples, including three small container functions from the Linux kernel.

For a class of extensions of lattice theory by unary function symbols and equational axioms (including modularity) algorithms for the computation of the corresponding initial algebra (if this is finite) are presented. In the case of finitely presented lattices the method amounts to a decision procedure for term equality and can be used for the computation of finite sublattices. Several examples and performance comparisons are provided.

I develop Alpay Algebra II, a self-contained formal framework that rigorously characterizes identity as an emergent fixed point in a categorical setting. Building only on Mac Lane's category-theoretic foundations and Bourbaki's structural paradigm, I define identity objects via unique solutions to self-referential functorial equations. In particular, given a transfinite endofunctor φ : A → A on a category A, I show that infinite data structures, streams, and symbolic transformations arise as unique fixed points of φ. The identity of a generative process is thereby identified with the initial (and universal) fixed point of φ, obtained as the limit of an ordinal-indexed iterative construction. All necessary definitions-categories, functors, algebras, universal morphisms, initial fixed points, and convergence of transfinite sequences-are provided within my formal development. I prove existence and uniqueness (up to isomorphism) of initial algebras (minimal fixed-point objects) under broad conditions, and I demonstrate that each such fixed point carries a universal property: it serves as the canonical representative of the process's identity. The main results establish that (1) every sufficiently continuous endofunctor admits a unique minimal fixed point (initial algebra) which is isomorphic to its own image under φ, and (2) this fixed point yields the intrinsic identity of the underlying generative system, in the sense that its associated structural morphism is an identity morphism in the emergent category of states. In sum, identity in Alpay Algebra is not an extra axiom but a necessary outcome of transfinite fixed-point convergence. The presentation is strictly mathematical and abstract: I employ categorical reasoning (initial objects, universal constructions, transfinite induction) without any reliance on external implementation or philosophical narrative. By focusing on structural convergence and universal properties, I illustrate the generality and necessity of this fixed-point characterization of identity. Minimal references to Bourbaki and Mac Lane contextualize my approach within the grand paradigm of structural foundations and category-theoretic logic.

We present a first-order theory of sequences with integer elements, Presburger arithmetic, and regular constraints, which can model significant properties of data structures such as arrays and lists. We give a decision procedure for the quantifier-free fragment, based on an encoding into the first-order theory of concatenation; the procedure has PSPACE complexity. The quantifier-free fragment of the theory of sequences can express properties such as sortedness and injectivity, as well as Boolean combinations of periodic and arithmetic facts relating the elements of the sequence and their positions (e.g., "for all even i's, the element at position i has value i+3 or 2i"). The resulting expressive power is orthogonal to that of the most expressive decidable logics for arrays. Some examples demonstrate that the fragment is also suitable to reason about sequence-manipulating programs within the standard framework of axiomatic semantics.

Synthesis of program fragments from specifications can make programs easier to write and easier to reason about. To integrate synthesis into programming languages, synthesis algorithms should behave in a predictable way - they should succeed for a well-defined class of specifications. They should also support unbounded data types such as numbers and data structures. We propose to generalize decision procedures into predictable and complete synthesis procedures. Such procedures are guaranteed to find code that satisfies the specification if such code exists. Moreover, we identify conditions under which synthesis will statically decide whether the solution is guaranteed to exist, and whether it is unique. We demonstrate our approach by starting from decision procedures for linear arithmetic and data structures and transforming them into synthesis procedures. We establish results on the size and the efficiency of the synthesized code. We show that such procedures are useful as a language extension with implicit value definitions, and we show how to extend a compiler to support such definitions. Our constructs provide the benefits of synthesis to programmers, without requiring them to learn new concepts or give up a deterministic execution model.

Semantic subtyping is an approach for defining sound and complete procedures to decide subtyping for expressive types, including union and intersection types; although it has been exploited especially in functional languages for XML based programming, recently it has been partially investigated in the context of object-oriented languages, and a sound and complete subtyping algorithm has been proposed for record types, but restricted to immutable fields, with union and recursive types interpreted coinductively to support cyclic objects. In this work we address the problem of studying semantic subtyping for imperative object-oriented languages, where fields can be mutable; in particular, we add read/write field annotations to record types, and, besides union, we consider intersection types as well, while maintaining coinductive interpretation of recursive types. In this way, we get a richer notion of type with a flexible subtyping relation, able to express a variety of type invariants useful for enforcing static guarantees for mutable objects. The addition of these features radically changes the defi- nition of subtyping, and, hence, the corresponding decision procedure, and surprisingly invalidates some subtyping laws that hold in the functional setting. We propose an intuitive model where mutable record val- ues contain type information to specify the values that can be correctly stored in fields. Such a model, and the correspond- ing subtyping rules, require particular care to avoid circularity between coinductive judgments and their negations which, by duality, have to be interpreted inductively. A sound and complete subtyping algorithm is provided, together with a prototype implementation.